Progressive addition power lens

ABSTRACT

A system and method for designing a progressive lens. Mean power is specified at points distributed over the entire surface of the lens and lens height is specified around the edge of the lens. Lens height is determined at the points consistent with the specified mean power and the lens edge height in part by solving a partial differential equation of the elliptic type subject to the lens edge height as a boundary condition. A successive over-relaxation technique may be employed to converge on the solution to the partial differential equation, and an over-relaxation factor may be determined to most efficiently relax the equation.

This application claims priority from international application numberPCT/GB02/02284 filed on May 31, 2002, the contents of which are herebyincorporated by reference in their entirety. This internationalapplication will be published under PCT Article 21(2) in English.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to progressive addition power ophthalmiclenses and, in particular, to an improved system and method fordesigning such lenses.

2. Description of the Related Art

Bifocal spectacle lenses have been used for many years by peoplesuffering from presbyopia, a medical condition that causes loss ofaccommodation of the eye with advancing age resulting in difficultyfocusing. Bifocal lenses provided a solution by dividing the lenseshorizontally into two regions, each having a different optical power.The upper region of the lens was designed with the appropriate opticalpower for distance viewing, while the lower region was designed forcloser viewing (e.g. reading). This allows the wearer to focus atdifferent distances by merely changing their gaze position. However,wearers frequently experienced discomfort due to the abrupt transitionbetween the different lens regions. As a consequence, progressiveaddition lenses were developed to provide a smooth transition in opticalpower between the regions of the lens.

Conventionally, progressive addition lenses are usually described ashaving three zones: an upper zone for far vision, a lower zone for nearvision, and an intermediate progression corridor that bridges the firsttwo zones. FIG. 1 is a diagram of a typical progressive lens shown invertical elevation (plan view). The lens has an distance zone 2 with agiven, relatively lower mean power and a reading zone 4 with relativelyhigher mean power. An intermediate progression corridor 6 of varying andusually increasing mean power connects the distance and reading zones.The outlying regions 8 adjoining the progression corridor and the lensboundary 10 (i.e. the edge of the lens) are also shown.

The goals in designing progressive lenses have been to provide bothessentially clear vision in upper and lower zones 2 and 4 and smoothvariation in optical power through the progression corridor 6, while atthe same time to control the distribution of astigmatism and otheroptical aberrations.

Early design techniques required the lens to be spherical throughout thedistance and reading zones, and employed various interpolative methodsto determine the lens shape in the progression corridor and outlyingregions. These techniques suffered from several disadvantages. Althoughthe optical properties of the distance zone, reading zone, andprogression corridor were usually satisfactory, regions adjoining theprogression corridor and lens edge tended to have significantastigmatism. Interpolative methods designed to compress astigmatism intoregions near the progression corridor yielded relatively steep gradientsin mean power, astigmatism and prism. The resulting visual field was notas smooth and continuous as would be desirable for comfort, ease offocusing, and maximizing the effective usable area of the lens.

FIG. 2 shows a three dimensional representation of the mean powerdistribution over the surface of a typical progressive lens design. Meanpower M is graphed in the vertical direction and the disc of the lens inshown against x and y coordinates. The disc of the lens is viewed froman angle less than 90° above the plane of the lens. The orientation ofthe lens is opposite of that in FIG. 1, the distance area with low meanpower 12 shown in the foreground of FIG. 2 and the reading area withhigh mean power 14 shown at the back. Steep gradients in mean power areevident, especially in the outlying regions 16.

Many progressive lens design systems permit the designer to set opticalproperties at only a few isolated points, curves, or zones of the lensand employ a variety of interpolative methods to determine the shape andoptical properties of the remainder of the lens.

U.S. Pat. No. 3,687,528 to Maitenaz, for example, describes a techniquein which the designer specifies the shape and optical properties of abase curve running from the upper part of the lens to its lower part.The base curve, or “meridian line” is the intersection of the lenssurface with the principal vertical meridian, a plane dividing the lensinto two symmetrical halves. The designer is constrained by therequirement that astigmatism vanish everywhere along the meridian line(i.e. the meridian line must be “umbilical”). Maitenaz discloses severalexplicit formulas for extrapolating the shape of the lens horizontallyfrom an umbilical meridian.

U.S. Pat. No. 4,315,673 to Guilino describes a method in which meanpower is specified along an umbilical meridian and provides an explicitformula for extrapolating the shape of the remainder of the lens.

In a Jul. 20, 1982 essay, “The TRUVISION® Progressive Power Lens,” J. T.Winthrop describes a progressive lens design method in which thedistance and reading zones are spherical. The design method describedincludes specifying mean power on the perimeters of the distance andreading zones, which are treated as the only boundaries.

U.S. Pat. No. 4,514,061 to Winthrop also describes a design system inwhich the distance and reading areas are spherical. The designerspecifies mean power in the distance and reading areas, as well as alongan umbilical meridian connecting the two areas. The shape of theremainder of the lens is determined by extrapolation along a set oflevel surfaces of a solution of the Laplace equation subject to boundaryconditions at the distance and reading areas but not at the edge of thelens. The lens designer cannot specify lens height directly at the edgeof the lens.

U.S. Pat. No. 4,861,153 to Winthrop also describes a system in which thedesigner specifies mean power along an umbilical meridian. Again, theshape of the remainder of the lens is determined by extrapolation alonga set of level surfaces of a solution of the Laplace equation thatintersect the umbilical meridian. No means is provided for the lensdesigner to specify lens height directly at the edge of the lens.

U.S. Pat. No. 4,606,622 to Furter and G. Furter, “Zeiss Gradal HS—Theprogressive addition lens with maximum wearing comfort”, ZeissInformation 97, 55-59, 1986, describe a method in which the lensdesigner specifies the mean power of the lens at a number of specialpoints in the progression corridor. The full surface shape is thenextrapolated using splines. The designer adjusts the mean power at thespecial points in order to improve the overall properties of thegenerated surface.

U.S. Pat. No. 5,886,766 to Kaga et al. describes a method in which thelens designer supplies only the “concept of the lens.” The designconcept includes specifications such as the mean power in the distancezone, the addition power, and an overall approximate shape of the lenssurface. Rather than being specified directly by the designer, thedistribution of mean power over the remainder of the lens surface issubsequently calculated.

U.S. Pat. No. 4,838,675 to Barkan et al. describes a method forimproving a progressive lens whose shape has already been roughlydescribed by a base surface function. An improved progressive lens iscalculated by selecting a function defined over some subregion of thelens, where the selected function is to be added to the base surfacefunction. The selected function is chosen from a family of functionsinterrelated by one or a few parameters; and the optimal selection ismade by extremizing the value of a predefined measure of merit.

In a system described by J. Loos, G. Greiner and H. P. Seidel, “Avariational approach to progressive lens design”, Computer Aided Design30, 595-602, 1998 and by M. Tazeroualti, “Designing a progressive lens”,in the book edited by P. J. Laurent et al., Curves and Surfaces inGeometric Design, A K Peters, 1994, pp. 467-474, the lens surface isdefined by a linear combination of spline functions. The coefficients ofthe spline functions are calculated to minimize the cost function. Thisdesign system does not impose boundary conditions on the surface, andtherefore lenses requiring a specific lens edge height profile cannot bedesigned using this method.

U.S. Pat. No. 6,302,540 to Katzman et al. discloses a lens design systemthat requires the designer to specify a curvature-dependent costfunction. In the Katzman system, the disk of the lens is preferablypartitioned into triangles. The system generates a lens surface shapethat is a linear combination of independent “shape polynomials,” ofwhich there are at least seven times as many as there are partitioningtriangles (8:17-40). The surface shape generated approximately minimizea cost function that depends nonlinearly on the coefficients of theshape polynomials (10:21-50). Calculating the coefficients requiresinverting repeatedly matrices of size equal to the number ofcoefficients. Since every shape polynomial contributes to the surfaceshape over every triangle, in general none of the matrices' elementsvanishes. As a result, inverting the matrices and calculating thecoefficients take time proportional to at least the second power of thenumber of shape polynomials.

The inherent inaccuracy of the shape polynomials (10:10-14) implies thatthe disk must be partitioned more finely wherever the mean power variesmore rapidly. These considerations set a lower limit on the number ofshape polynomial coefficients that would have to be calculated, andhence the time the system would need to calculate the lens surfaceshape. Since the Katzman system requires time that is at least quadraticin the number of triangles to calculate the lens surface, the system isinherently too slow to return a calculated lens surface to the designerquickly enough for the designer to work interactively with the system.The inherent processing delay prevents the designer from being able tocreate a lens design and then make adjustments to the design whileobserving the effects of the adjustments in real-time.

None of the above design systems provides a simple method by which thelens designer can specify the desired optical properties over the entiresurface of the lens and derive a design consistent with those opticalproperties. As a consequence, many of these prior systems result inoptical defects in the outlying regions of the lens and unnecessarilysteep gradients in mean power. Furthermore, the computational complexityof some of the prior systems result in a lengthy design process thatdoes not permit the lens designer to design the lens interactively. Manyof the prior systems also do not include a definition of the lens heightaround the periphery of the lens and therefore do not maximize theuseful area of the lens.

BRIEF SUMMARY OF THE INVENTION

The present invention seeks to provide the lens designer with a means tospecify as parameters both the mean power of the lens over its entiresurface and the height of the lens around its boundary and to obtain thesurface shape of the lens consistent with those parameters in time shortenough for the designer to make use of interactively. Lens designs canbe created which have smooth, continuous optical properties desirablefor wearer comfort, ease of adaptation, and maximizing the effectiveusable area of the lens.

The present invention differs from previous design processes whichgenerally start by modeling the lens surface shape directly, calculatingoptical properties, and then attempting to modify the surface shape soas to optimize optical properties. The prior art process of varyingsurface shape to achieve desired optical properties is unstablenumerically. For this reason, previous design processes cannot be reliedupon to generate lens designs quickly enough for the designer to useinteractively. In contrast to previous design processes, the presentinvention starts with a prescription of the key optical property of meanpower over the lens surface, together with the lens edge height, andthen calculates the lens surface shape.

In accordance with the invention, mean power is specified at a pluralityof points distributed over the entire surface of the lens and lensheight is specified around the edge of the lens. Lens height isdetermined at the plurality of points consistent with the specified meanpower and the lens edge height in part by finding a unique solution to apartial differential equation of the elliptic type subject to the lensedge height as a boundary condition.

The present invention preferably incorporates a method forredistributing astigmatism in the lens design. The method redistributesastigmatism more evenly over the surface of the lens and reduces peaksof astigmatism in critical areas. The present invention preferably alsoincorporates a method of creating special lens designs for left andright eyes whilst maintaining horizontal symmetry and prism balance.

The method of the present invention is preferably implemented usingsoftware executing on a computer to provide a system to define a lenssurface shape in an interactive manner with smooth, continuous opticalproperties desirable for wearer comfort, ease of adaptation, and maximumeffective use of the lens area.

The invention also comprises a progressive lens designed according tothe disclosed design method. Preferred embodiments of a lens include aprogressive lens having a distance area and a reading area wherein meanpower over the lens surface varies according to a set of curves formingiso-mean power contours on the lens surface and a contour defining anarea of constant mean power in the distance area is an ellipse with aratio of major axis to minor axis in the range of about 1.1 to 3.0.Another preferred embodiment of a lens includes a distance area having afirst mean power, a reading area having a second mean power higher thanthe first mean power, and a central area between the distance andreading areas with a width of at least about 10 millimeters wide and inwhich mean power increases smoothly and substantially monotonicallythroughout the central area in a direction from the distance area to thereading area.

The invention also comprises a system for designing progressive lenscomprising a processor for accepting inputs defining mean powervariation over a coordinate system covering the surface of the lens anddefining lens height around the edge of the lens and for calculatinglens height at a plurality of points over the lens surface by solving anelliptic partial differential equation subject to the lens height at theedge of the lens as a boundary condition, and a memory for storing thecalculated lens height values. The lens design created using the systemof the present invention is preferably manufactured using a CNCcontrolled grinding or milling machine using techniques well known inthe art.

Further aspects of the invention are described hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

The following is a description of certain embodiments of the invention,given by way of example only and with reference to the followingdrawings, in which:

FIG. 1 is a diagram of a conventional progressive lens, shown invertical elevation.

FIG. 2 is a three dimensional representation of the mean powerdistribution over the surface of a typical prior art progressive lens.

FIG. 3 is a diagram of a lens according to an embodiment of the currentinvention in vertical elevation view showing a connecting path and arepresentative subset of a system of contours intersecting theconnecting path.

FIG. 4 is a graph showing an example of a function specifying mean poweralong the connecting path.

FIG. 5 is a vertical elevation view of the surface of a lens showing apreferred coordinate system comprising x- and y-axes and angle θ.

FIG. 6 is a vertical elevation view of the surface of a lens accordingto an embodiment of the current invention showing the boundary areas ofthe lens.

FIG. 7 is a graph showing an example of a lens boundary height functionranging from θ=0 to 360 degrees around the lens boundary.

FIG. 8 is a three dimensional representation of a theoretical mean powerdistribution over the surface of a lens according to an embodiment ofthe present invention.

FIG. 9 is a graph showing an example of optimizing the mean powerprofile along the connecting path.

FIG. 10 is a vertical elevation view of the surface of a lens showing anexample of mean power distribution over a family of iso-mean powerellipses.

FIG. 11 is a graph showing an example of an edge height profile aroundthe periphery of a lens.

FIG. 12 is a vertical elevation view of the surface of a lens showing anexample of the distribution of astigmatism resulting from the mean powerdistribution of FIG. 10 and edge height profile of FIG. 11.

FIG. 13 is a vertical elevation view of the surface of a lens showing anexample of altered mean power distribution to reduce astigmatism alongthe centerline.

FIG. 14 is a graph showing an example of an altered edge height profile.

FIG. 15 is a vertical elevation view of the surface of a lens showing anexample of the redistribution of astigmatism resulting from the alteredmean power distribution of FIG. 13 and altered edge height profile ofFIG. 14.

FIG. 16 is a vertical elevation view of the surface of a lens showing anexample of a mean power distribution incorporating the change in meanpower profile along the connecting path as shown in FIG. 17.

FIG. 17 is a graph showing an example of change in the mean powerprofile along the connecting path to optimize the mean power in thecentral corridor area.

FIG. 18 is a vertical elevation view of the surface of a lens showing anexample of an astigmatism distribution derived from the recalculatedsurface height distribution.

FIG. 19 is a vertical elevation view of the surface of a lens showing anexample of a rotated mean power distribution.

FIG. 20 is a graph showing an example of a rotated edge height profile.

FIG. 21 is a vertical elevation view of the surface of a lens showing anexample of astigmatism distribution resulting from the rotated meanpower distribution of FIG. 19 and rotated edge height profile of FIG.20.

FIG. 22 is a flowchart showing the major steps of one embodiment of thedesign method of the present invention.

DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Certain embodiments of the invention will now be described, by way ofexample only, to illustrate the subject matter of the invention. Thesurface of a lens may be described by the equation z=f(x, y), where x,y, and z are rectangular Cartesian co-ordinates.

For brevity let${\partial_{x}{\equiv \frac{\partial}{\partial x}}};{\partial_{y}{\equiv \frac{\partial}{\partial y}}};{\partial_{x}^{2}{\equiv \frac{\partial^{2}}{\partial x^{2}}}};{\partial_{y}^{2}{\equiv \frac{\partial^{2}}{\partial y^{2}}}};{{{and}\quad \partial_{xy}^{2}} \equiv {\frac{\partial^{2}}{{\partial x}{\partial y}}.}}$

The principal radii of curvature R₁ and R₂ of the surface are the rootsof the quadratic equation:

[rt−s ² ]R ² +h[2pqs−(1+p ²)t−(1+q ²)r]R+h ⁴=0  (1)

where p≡∂_(x)z, q≡∂_(y)z, r≡∂_(x) ²z, s≡∂_(xy) ²z, t≡∂_(y) ²z and$h \equiv {\sqrt{\left( {1 + p^{2} + q^{2}} \right)}.}$

See, e.g., I. N. Bronshtein & K. A. Semendyayev, “A Guide Book toMathematics,” Verlag Harri Deutsch, 1971, hereby incorporated byreference in its entirety.

The principal values of curvature are 1/R₁ and 1/R₂ respectively. Theprincipal curvature difference${\langle\delta\rangle} \equiv {{\frac{1}{R_{1}} - \frac{1}{R_{2}}}}$

is related to the optical property of astigmatism (also known ascylinder power) by D=1000(n−1)<δ> where D is measured in diopters, n isthe refractive index, and distance is measured in millimeters.

The mean curvature${\langle\mu\rangle} \equiv {\frac{1}{2}\left( {\frac{1}{R_{1}} + \frac{1}{R_{2}}} \right)}$

is similarly related to the optical property of mean powerM=1000(n−1)<μ> also measured in diopters. As used herein, <μ> is themean of the two principal curvatures, and <δ> is the absolute differenceof the two principal curvatures.

In one embodiment of the present invention, the designer preferablyprescribes M(x, y) and thus <μ>(x,y) over the entire lens area. Fordesigning a progressive lens, mean power is prescribed over the entirelens surface using a preferred system of coordinates. This preferredsystem consists of a continuous set of non-mutually intersectingcontours that collectively fill the entire area of the lens and aconnecting path, with each contour line intersecting the connecting pathonce. The connecting path is a curve connecting a point in the distancearea to one in the reading area. To specify the mean power in thispreferred system of coordinates, the designer specifies how mean powervaries along the connecting path and how mean power varies along eachcontour from its point of intersection with the connecting path.Preferably, variation in mean power along the connecting path should bedescribed by a suitably smooth function ranging from a lower value inthe distance area to a higher value in the reading area.

It is also preferred that lens height around the boundary be specifiedby a function that varies little near the distance and reading areas andgradually near the intermediate regions shown in FIG. 6. One way toconstruct such a function is as a smooth, piecewise composite of any ofa wide variety of well-known elementary functions, such as polynomial,trigonometric, or gaussian functions.

The lens surface shape is then determined on the basis of the mean powerdistribution and lens height at the boundary. The preferred method is tosolve a boundary value problem.

Unwanted astigmatism in critical areas may then be reduced andindividual left and right lens designs may be created, as described inmore detail below.

A. Prescribing Mean Power Over the Lens Surface

For a progressive lens of the present invention, a preferred method forprescribing the mean power M as a function over the entire lens areainvolves four steps. First, the designer selects points, P_(D) in thedistance area and P_(R) in the reading area, and a path connecting thosepoints. In one embodiment, both of the points and the connecting pathlie along the left-right symmetry axis of the lens. Therefore in thisembodiment the connecting path is referred to as the power profilemeridian. FIG. 3 is a vertical elevation (plan view) of a progressivelens showing the selected points P_(D) and P_(R) shown as endpoints 20and 22 at each end of connecting path (or power profile meridian) 24.

Second, a continuous set of contours is selected, subject to theconditions that each contour in the set intersects the power profilemeridian once and no two contours in the set intersect one another. Thecurves in FIG. 3 within the lens boundary 10 are representative membersof one example of such a continuous set of contours. In one preferredembodiment, the set of contours collectively fill the entire disk of thelens. In a second preferred embodiment, the designer may define adistance area 32 and reading area 34 of constant mean power. The set ofcontours collectively fill the remaining lens area. In the example shownin FIG. 3, curves 28 and 30 are contours that form the boundaries of theareas of constant mean power. In this example, the continuous set ofcontours consists of two families of hyperbolae: $\begin{matrix}{{\frac{x^{2}}{\xi_{R}^{2}} - \frac{\left( {y - P_{R}} \right)^{2}}{_{R}^{2}}} = {{1\quad {for}\quad y} \leq {0\quad {and}}}} & \left( {2A} \right)\end{matrix}$

$\begin{matrix}{{{{\frac{x^{2}}{\xi_{D}^{2}} - \frac{\left( {y - P_{D}} \right)^{2}}{_{D}^{2}}} = {{1\quad {for}\quad y} \geq 0}},}\quad} & \left( {2B} \right)\end{matrix}$

where the x and y coordinates are defined according the coordinatesystem as shown in FIG. 5. As the parameters ζ_(r) and ζ_(D) vary, theset of contours fill the entire area between contours 28 and 30. Fory=0, the two families of hyperbolae overlap, each including the equator26 of the disk as a member when ξ_(D) or ξ_(R) is varied.

The set of contour lines illustrated in FIG. 3 is by no means an uniqueexample of contours that can fulfill the conditions given above.Contours may be selected from families of curves other than conics, andfrom families of conics other than hyperbolae. In the second preferredembodiment, the set of contours could equally well consist of twofamilies of ellipses. In one example of this embodiment, the contourforming the boundary of the distance area is preferably an ellipse witha ratio of major axis to minor axis in the range of about 1.1 to 3.0. Itis also expected that contours may be selected from families of curvesother than conics.

In a third step of a preferred method for prescribing the mean powerover the lens, the designer prescribes a function specifying thevariation in mean power along the power profile meridian. Preferably,the prescribed function takes into account criteria of wearer comfortand the intended use of the lens. Functions that meet such criteria may,for example, be linear combinations of elementary functions. One exampleof such a function is:${M(y)} = {M_{D} + {\left\lbrack \frac{M_{R} - M_{D}}{2} \right\rbrack \left\lbrack {1 - {\cos \left( {\pi \frac{y_{D} - y}{y_{D} - y_{R}}} \right)}} \right\rbrack}}$

where M_(D) is mean power specified at a point P_(D)=(0,y_(D)) in thedistance area and M_(R) is mean power specified at a pointP_(R)=(0,y_(R)) in the reading area.

FIG. 4 is a graph of mean power M plotted against the y-axis of the lensalong the length of the power profile meridian. The ends of the graphcorrespond to the endpoints 20 and 22 of the power profile meridian. Afunction 36 specifying mean power M along the power profile meridian isshown as an example of a suitable variation of mean power. In theexample shown, mean power is constant in the distance area 32 andreading area 34.

Finally, in the fourth step of a preferred method for prescribing themean power over the lens, the designer prescribes functions specifyingthe variation in mean power M along each of the contour lines. The meanpower at the point where a contour intersects the power profile meridianequals the mean power specified at that point on the power profilemeridian. Thus, defining the variation in mean power M along each of thecontour lines completes the definition of mean power over the entiresurface of the lens. One convenient choice consistent with thisrequirement is for mean power simply to remain constant along eachcontour. Other choices are also compatible with the disclosedembodiment.

B. Prescribing Lens Height at the Lens Boundary

In a preferred embodiment, the designer also prescribes the lens heightat the edge of the lens. (As used herein, the terms “lens edge” and“lens boundary” are synonymous). The designer specifies a lens boundaryheight function z(θ) where z denotes the height of the lens and θdenotes the angular coordinate around the boundary of the lens. FIG. 5illustrates the preferred convention for θ to be defined as the anglearound the edge 48 of the lens in an anticlockwise direction starting atthe x-intercept of the edge of the lens.

Preferably, the designer's specification of z(θ) takes into accountcriteria of wearer comfort and the intended use of the lens.Discontinuities or abrupt changes in z(θ) generally lead to image jumpsthat are uncomfortable for the wearer. Also, to be supported in aneyeglass frame a lens should be neither too thick nor too thin aroundits edge.

For a progressive lens, additional design criteria preferably apply toz(θ). FIG. 6, a vertical elevation (plan view) of the surface of aprogressive lens, illustrates segments of the lens boundary thatcorrespond roughly to adjoining areas of a typical progressive lens,illustrated in FIG. 1. Boundary segment 50 roughly adjoins distance area40;

boundary segment 52 roughly adjoins reading area 42; and the boundarysegments 54 and 56 roughly adjoin the outlying regions 44 and 46. Tofacilitate designs with relatively uniform optical properties in thedistance and reading areas, it is preferred that z(θ) vary little withineach of segments 50 and 52. To facilitate designs that do not produceuncomfortable image distortions at the lens periphery, it is preferredthat in segments 54 and 56, z(θ) vary gradually so as to makesubstantially smooth transitions between segments 50 and 52. In order tofulfill these design criteria the designer may, for example, constructz(θ) from a smooth, piecewise composite of any of a wide variety ofwell-known elementary functions, such as polynomial or trigonometricfunctions.

FIG. 7 illustrates the preferred qualitative behavior of the lensboundary height function 60, showing lens boundary height z on thevertical axis plotted against angular coordinate θ on the horizontalaxis, with θ varying from 0 to 360 degrees over lens boundary segments50, 52, 54, and 56.

Within these criteria, some flexibility remains in the specification ofz(θ) for a progressive lens. After examining the optical properties of alens whose surface shape has been determined according to the presentembodiment, a designer may exploit this flexibility by modifying z(θ).It has been shown that a typical progressive lens can be designed andoptimized using the methods described herein in an hour or less, witheach successive calculation of lens height distribution over the surfaceof the lens being performed in a matter of minutes. With the benefit ofsuch rapid feedback, the designer may modify z(θ) in a way that leads toimproved optical properties, such as lowered astigmatism, in criticalareas of the lens.

C. Determining the Lens Surface Shape

When the mean curvature function <μ> is specified, the height functionsatisfies: $\begin{matrix}{{\left\lbrack {\partial_{x}^{2}{+ \partial_{y}^{2}}} \right\rbrack z} = F} & (3)\end{matrix}$

where

F=2<μ>[1+(∂_(x) z)²+(∂_(y) z)²]^(3/2)−[∂_(y) z] ²∂_(x) ² z+2[(∂_(x)z)(∂_(y) z)]∂_(xy) ² z−[∂ _(x) z] ²∂_(y) ² z

The present embodiment determines the lens surface shape by solvingequation (3) subject to the boundary condition that the lens edge heightz(θ) is specified. Since equation (3) is a partial differential equationof the elliptic type, a unique, stable solution for the lens surfaceshape z must exist. To determine that solution, the present embodimentuses an iterative process to solve equation (3) numerically.

To establish a starting solution, a low power lens configuration, inwhich |∂_(x)z|<<1 and |∂_(y)z|<<1, is assumed. For this startingsolution, ∂_(x)z, and ∂_(y)z may be eliminated from equation (3),leading to a Poisson equation: $\begin{matrix}{{\left\lbrack {\partial_{x}^{2}{+ \partial_{y}^{2}}} \right\rbrack z} = {2{\langle\mu\rangle}}} & (4)\end{matrix}$

The area of the lens is covered with a square mesh. Mean curvature <μ>is evaluated from M at mesh points; its values are proportional to F⁽⁰⁾,a function of two discrete variables. Throughout, a function with asuperscript in parentheses will be the function of two discretevariables representing the values at mesh points of the correspondingcontinuum function. z⁽⁰⁾ at mesh points on the boundary of the diskrepresents values of z(θ) at those points. For mesh points near theboundary of the disk, the value of z⁽⁰⁾ is a suitable average of nearbyvalues of z(θ). Elsewhere on the mesh, z⁽⁰⁾ need not be defined.

Superscripts in parentheses refer to the stage of the iteration. In thefirst iteration, the discrete analog of equation (4): $\begin{matrix}{{\left( {\partial_{x}^{2}z} \right)^{(1)} + \left( {\partial_{y}^{2}z} \right)^{(1)}} = {{F^{(0)}\quad {with}\quad F^{(0)}} = {2{\langle\mu\rangle}}}} & (5)\end{matrix}$

is solved to obtain z⁽¹⁾. z⁽¹⁾ is solved for at mesh points using theSuccessive Over-Relaxation (SOR) Technique. The SOR technique forsolving elliptic equations is discussed in W. H. Press et al.,“Numerical Recipes in C: The Art of Scientific Computing” (CambridgeUniversity Press 1992) at sections 19.2 and 19.5, which are herebyincorporated by reference.

In subsequent iterations, equation (6), the discrete analog of equation(3) $\begin{matrix}{{\left( {\partial_{x}^{2}\quad z} \right)^{({n + 1})} + \left( {\partial_{y}^{2}\quad z} \right)^{({n + 1})}} = F^{(n)}} & (6)\end{matrix}$

is solved to obtain z^((n+1)). For n≧1, F^((n)) includes all the termsshown in equation (3).

The values of F^((n)) are calculated at mesh points using the values ofz^((n)) determined at mesh points in the previous iteration. The partialdifferentials of z that appear in F, as shown in equation (3), arecalculated using central difference schemes, with special care beingtaken for mesh points near the circular boundary. Again, the SORtechnique is used to solve for z^((n+1)) at mesh points.

The SOR technique employs a repetitive series of sweeps over the mesh toconverge on a solution. The rate of convergence is dependent on thevalue of the Over-Relaxation Factor (ORF), and a preferred value of theORF is determined experimentally. Once determined, the same ORF value isalso preferred for solving similar equations, such as successiveiterations of equation (6). (See Press et al., at section 19.5)

An important advantage of the SOR technique is that it reachesconvergence in a time proportional to the square root of the number ofmesh points. This feature implies that at modest cost in computationaltime, a sufficient mesh density can be implemented for SOR to convergeto the solution of equation (6) that corresponds at mesh points to theunique solution of equation (3).

It has been found that five iterations of equation (6) will typicallyproduce a satisfactory numerical solution of equation (3).

D. Reducing Unwanted Astigmatism in Critical Areas

Using the lens surface shape resulting from step C above, principalcurvature difference <δ> can be calculated at every mesh point. Partialdifferentials of z that appear in <δ> are calculated using centraldifference schemes, with special care being taken for mesh points nearthe circular boundary.

Excessive astigmatism may be found in critical areas such as the centraland reading areas. While astigmatism cannot be avoided entirely in aprogressive lens design, astigmatism can be redistributed more evenly,away from critical areas.

Astigmatism in the central area, for example, can be reduced to improveoptical performance there. A criterion for the maximum level ofastigmatism acceptable in the central area, such asD≦0.15*(M_(R)−M_(D)), could be imposed. Here, M_(D) is mean powerspecified at point P_(D) in the distance area and M_(R) is mean powerspecified at point P_(R) in the reading area.

Assume that the lens shape is symmetrical about the centerline, so thatz=f(x, y) with f(−x, y)=f(x, y). Then along the centerline, p=0 and s=0and the mean curvature <μ> and principal curvature difference <δ> arerespectively given by equation (7): $\begin{matrix}{\begin{matrix}{{\langle\mu\rangle} = \frac{\left( {t + {h^{2}r}} \right)}{2h^{3}}} & \quad & {and} & \quad & {{\langle\delta\rangle} = \frac{\left( {t - {h^{2}r}} \right)}{h^{3}}}\end{matrix}\quad.} & (7)\end{matrix}$

To make astigmatism D vanish exactly along the centerline, it would benecessary for t to be made equal to h²r and mean curvature <μ> madeequal to r/h. Therefore the <μ> function would have to be modifiedaccording to equation (8):

<μ>(0,y)→<μ>(0,y)+Δ<μ>(0,y)  (8)

where $\begin{matrix}{\left. {{\Delta {\langle\mu\rangle}\left( {0,y} \right)} \equiv \frac{r}{h}} \middle| {}_{({0,y})}{{- {\langle\mu\rangle}}\left( {0,y} \right)} \right. = \left. \frac{\partial_{x}^{2}\quad z}{\sqrt{1 + \left( {\partial_{y}\quad z} \right)^{2}}} \middle| {}_{({0,h})}{{- {\langle\mu\rangle}}\left( {0,y} \right)} \right.} & (9)\end{matrix}$

To reduce astigmatism D in the central area and at the same timedistribute changes in mean power M across the lens, a spreading functionσ(x) can be employed:

<μ>(x,y)→<μ>(x,y)+σ(x)Δ<μ>(0,y)  (10)

σ(x) may be any smoothly-varying function that takes the value 1 at x=0.One example of such a function is: $\begin{matrix}{{\sigma (x)} = \begin{Bmatrix}{\exp \left( {- {k^{2}\left( {x - x_{L}} \right)}^{2}} \right)} & \left| {x < x_{L}} \right. \\{\exp \left( {- {k^{2}\left( {x - x_{R}} \right)}^{2}} \right)} & \left| {x > x_{R}} \right. \\1 & \left| {x_{L} \leq x \leq x_{R}} \right.\end{Bmatrix}} & (11)\end{matrix}$

where x_(R) and −x_(L) take equal values prior to handing and create aconstant region for the spreading function σ(x). The parameter kcontrols the rate of decay of σ(x) to the left and to the right of theconstant region. The mean curvature function <μ>(x,y) resulting fromequations (8), (9) and (10) can be calculated at mesh points and is usedto completely recalculate the surface height function z in the mannerdescribed in step C.

In the complete recalculation of z with a selected σ(x), the derivativesinvolved in equation (9) will of course in general adopt new values. Asa result, the mean curvature function <μ>(x,y) will also adopt newvalues. The variables z, <μ>, and <δ> are recalculated repeatedly, withrepetition halting at the discretion of the designer. If necessary thevalues of x_(L), x_(R), and k can themselves be changed during thisprocess.

Astigmatism can be reduced similarly in any critical area, first bydetermining the local change in M required to make D vanish exactly inthe area, and then by distributing the change in M across the lens. Theresult is a set of modified values of M at mesh points. The modified Mis plugged back into step C at equation (5) to obtain a modified lenssurface function z at mesh points. The modified z, in turn, is used torecalculate the astigmatism D at mesh points, and the whole process maybe repeated until the distribution of astigmatism is found acceptable.

E. Optimizing the Mean Power Distribution Around the Power ProfileMeridian

As a result of changing the mean power to reduce unwanted astigmatism,it may be found that the mean power in certain critical areas is nolonger what the designer desires. FIG. 9 shows an example of a meanpower profile after reducing the astigmatism (line 72). For a typicaldesign, it will be desired to maintain the mean power below a certainvalue at the fit point (line 74). It will also be desired for the meanpower to reach the correct addition power, in this example 2.00diopters, at the addition measuring point (line 76). To achieve thedesired mean power profile, without raising the astigmatism levelssignificantly, the mean power can be altered locally. This is shown byline 70 in FIG. 9, and this alteration is made over some limited widthin the x direction of, for example, 12 to 16 mm. The modified mean canbe distributed across the lens in a simple linear fashion. The new Mdistribution is plugged back into step C at equation (5) to obtain amodified lens surface function z at mesh points. The modified z, inturn, is used to recalculate the astigmatism D at mesh points, so thatit can be checked to be within acceptable limits. The whole process maybe repeated until the distribution of mean and astigmatism is foundacceptable.

F. Designing Left and Right Lenses

Once an acceptable lens shape has been obtained, right-hand andleft-hand versions are designed in order to minimize binocularimbalance. Contrary to previous approaches to the handing problem, thedirect control of handing mechanics resides in the mean curvature andedge height prescriptions. In order to accomplish this handed lenses aredesigned by rotating both the mean curvature <μ>(x,y) and the boundaryheight z(θ) in an angle-dependent manner. Specifically,

<μ>(ρ,θ)→<μ>(ρ,H(θ))  (12)

z(θ)→z(H(θ))  (13)

where (ρ,θ) are polar coordinates corresponding to (x, y). The handingfunction H is of the form: $\begin{matrix}{{H(\theta)} = {h_{0}\begin{Bmatrix}{\exp \left( {- {K^{2}\left( {\theta - \frac{3\quad \pi}{2} + \omega} \right)}^{2}} \right)} & \left| {\theta < {\frac{3\quad \pi}{2} - \omega}} \right. \\{\exp \left( {- {K^{2}\left( {\theta - \frac{3\quad \pi}{2} - \omega} \right)}^{2}} \right)} & \left| {\theta > {\frac{3\pi}{2} + \omega}} \right. \\1 & \left| {{\frac{3\quad \pi}{2} - \omega} \leq \theta \leq {\frac{3\quad \pi}{2} + \omega}} \right.\end{Bmatrix}}} & (14)\end{matrix}$

where h₀ is the handing angle, ω controls the undistorted portion of thehanded reading region, and K determines the nature of the regions aheadof and behind the pure rotation. Typical values of these parameterscould be h₀=9 degrees, ω=30 degrees, and K=1.5. The mean curvatures andedge height values are plugged into step C at equation (5) to obtain arecalculated lens surface function z(x, y) at mesh points.

FIG. 8 is a three dimensional representation of a theoretical meancurvature distribution over the surface of a lens according to anembodiment of the current invention. Mean curvature M is graphed in thevertical direction. M is graphed as a function of x and y, which areshown as the two horizontal directions. The disc of the lens is viewedfrom an angle less than 90° above the plane of the lens. Since thedistance area is shown in the foreground and the reading area at thebackground of FIG. 8, y increases in the background-to-foregrounddirection. As can be seen, there is a region of lesser mean curvature 62in the distance area and greater mean curvature 64 in the reading area.The mean power transitions smoothly and increases substantiallymonotonically with increasing y throughout the optically critical areabetween the distance area and the reading area as well as in theoutlying areas 66 and 68.

G. An Example Lens Design

The following is an example of a lens design produced using methodscomprising the present invention. A mean power distribution is initiallydefined for the complete surface of the lens. A suitable distributionusing a family of iso-mean power ellipses can be seen in FIG. 10, inwhich contour lines are shown having mean power values between 0.25 to2.00 diopters in increments of 0.25 diopters. To completely define theentire surface it is also necessary to specify the lens height aroundthe edge of the lens. An example of a suitable lens edge height functionis shown in FIG. 11. This figure shows the lens surface height z inmillimeters referenced from the edge of the distance area. Theseparameters are used as inputs for equation (10), discussed above in stepC, and the equations are solved for the surface heights z over thecomplete surface. The solution is derived numerically using a high-speeddigital computer, using software and programming techniques well knownin the art. A suitable machine would be a personal computer with aPentium III or later processor, such as a Compaq EVO D300. Thecomputation time required to solve the boundary value problem isapproximately proportional to the square root of the number of points atwhich the height is calculated.

From the resultant z height values, the distribution of astigmatism andsphere power can be calculated for the design. Although the distributionof sphere power can in principle be calculated directly from the definedmean power distribution and the calculated distribution of astigmatism,calculating the distribution of sphere power from the resultant z heightvalues is useful to confirm that those z height values are consistentwith the defined mean power distribution. FIG. 12 shows the distributionof astigmatism resulting from the mean power distribution and lens edgeheight function of FIG. 10 and FIG. 11.

The next step is to reduce the astigmatism in the central corridor areato an acceptable level. This reduction is achieved by alteration in themean power distribution according to equations (8), (9) and (10)described above. The resulting altered mean power distribution is shownin FIG. 13. To take criteria of patient comfort into account as ispreferred, the lens edge height function may also be altered, and anexample of an altered lens edge height function is shown in FIG. 14. Thealtered mean power and edge height function are then used to recalculatethe distribution of surface heights z over the complete surface bysolving equation (5) as before. An astigmatism distribution derived fromthe recalculated surface height distribution is shown in FIG. 15. Thisstep may be repeated until the designer finds the astigmatismdistribution acceptable.

An example of the changes caused in the mean power profile along thecenterline to reduce astigmatism in the central corridor area is shownin FIG. 17. Having found that the astigmatism in the central corridorarea has been reduced to acceptable levels, the designer may find thatthe mean power profile along the centerline no longer complies with whatwas originally desired. As shown in FIG. 17 for example, in the additionarea, the mean power at the addition measurement point (−13 mm) is belowthe desired 2.00 diopters. Then the mean power profile must be optimizedaccording to step E above. The optimized mean power profile over a 12 mmwidth surrounding the centerline of the lens is then used as input torecalculate the distribution of surface heights z by again solvingequation (5). A mean power distribution incorporating the changes shownin FIG. 17, is shown in FIG. 16. The altered mean power and the previousedge height profiles function are then used to recalculate thedistribution of surface heights z over the complete surface by solvingequation (5) as before and an astigmatism distribution derived from therecalculated surface height distribution is shown in FIG. 18.

Finally the design has to be handed for use in a left or right eye of aspectacle frame. This is achieved by rotating both the mean powerdistribution and lens edge height function as described in step F aboveand once again recalculating the distribution of surface heights z bysolving equation (5) described above. An example of a rotated mean powerdistribution is shown in FIG. 19 and an example of a rotated edge heightfunction in shown in FIG. 20. After recalculating surface heights z, theastigmatism distribution is again derived. An example of such anastigmatism distribution can be seen in FIG. 21.

As can be seen by FIG. 19, the completed lens design includes a distancearea with relatively lower mean power in the top part of the lens and areading area with relatively higher mean power in the bottom part of thelens. Throughout a central area that extends between the distance andreading areas, mean power increases smoothly and substantiallymonotonically in the direction from the distance area to the readingarea. In a preferred embodiment, this central area is at least 30millimeters wide, but may vary about this width according to the lensdesign. In some designs the minimum width of the central portion may beabout 20 millimeters wide, or may be about 10 millimeters wide.

The resulting distribution of surface heights z can then be used in anyof the following ways:

a. To directly machine the progressive surface onto a plastic or glasslens;

b. To directly machine a glass or metal mould which will be used toproduce a progressive plastic lens by either casting or molding; or

c. To machine a ceramic former in either a convex form which will beused to produce a glass progressive lens by a slumping process, or aconcave form which will be used to produce a glass mould by a slumpingprocess from which a plastic progressive lens can be cast.

As discussed above, the calculation of the surface heights z ispreferably performed on a computer. The resulting data representing thedistribution of surface heights is preferably stored in the computer'smemory, and may be saved to a hard disk drive, CD-ROM, magnetic tape, orother suitable recording medium.

The machining is preferably performed by electronically transmitting thesurface height data to a computer numerically-controlled (CNC) millingor grinding machine. Examples of suitable CNC machines include aSchneider HSC 100 CNC to directly machine the progressive surface onto aplastic or glass lens, a Mikron VCP600 to machine a glass or metalmould, and a Mikron WF32C or Schneider HSG 100 CNC to machine a ceramicformer, although other suitable machines are well known to those ofskill in the art.

In each of the above cases the distribution of surface heights z must bepost-processed to suit the particular CNC controller on thegrinding/milling machine used. Compensation must also be built in to thesurface geometry depending on the size and type of grinding tool/cutterused to ensure that the design surface is produced. In the case ofmachining ceramic formers for use with a slumping process, furthercompensation must be built into the distribution of surface heights z totake care of unwanted geometry changes. These result from the bendingand flowing of the glass as it is heated up to its softeningtemperature, to allow it to take up the shape of the ceramic former.

Lenses produced according to the present invention need not have acircular outline. As part of any of the above manufacturing procedures,lenses may be glazed into a variety of outlines for a variety ofspectacle frames. Furthermore, the lens edge height used in thecalculation of lens surface heights z need not be the physical edge ofthe lens blank. For example, a typical 70 millimeter circular lens blankmay have edge heights defined 10 millimeters in from the actual edge ofthe lens blank, depending on the size of the lens ultimately required.In this example, the designer's specification of mean power and thecalculation of lens surface heights z will apply for the lens areawithin the boundary at which lens edge height is defined, rather thanfor the entire surface of the lens blank.

A flow chart showing the process described above is provided in FIG. 22.The flowchart illustrates each of the main steps involved in the designand manufacturing process for a progressive lens as described above. Itshould be noted that FIG. 22 describes only one example of a design andmanufacturing process and not all of the steps shown in the flow chartmay be necessary for a given lens design.

Thus an improved method for designing progressive addition powerophthalmic lenses has been described. It will be appreciated that themethod has been described in terms of several embodiments, which aresusceptible to various modifications and alternative forms. Accordingly,although specific embodiments have been described, these are examplesonly and are not limiting upon the scope of the invention.

What is claimed is:
 1. A method for designing a progressive lens surfacecomprising: specifying mean power at a plurality of points distributedover the entire surface of the lens; specifying lens height around theedge of the lens; and determining lens height at the plurality of pointsconsistent with the specified mean power and lens edge height,comprising finding a unique solution of a partial differential equationof the elliptic type subject to a boundary condition of the lens edgeheight.
 2. The method of claim 1, wherein finding the unique solution ofthe partial differential equation comprises: employing a successiveover-relaxation technique to converge on the solution; and determiningan over-relaxation factor to most efficiently relax the equation.
 3. Themethod of claim 2, wherein the step of determining the lens height at aplurality of points comprises: defining a mesh comprising a plurality ofpoints over the surface of the lens; determining mean power at eachpoint on the mesh as defined by the specified mean power distributionover the lens surface; and numerically solving on the mesh a partialdifferential equation of the elliptic type, subject to the lens edgeheight as a boundary condition, to determine the lens height at eachpoint of the mesh.
 4. The method of claim 3, wherein the lens comprisesa distance area and a reading area, and wherein the step of specifyingmean power further comprises: specifying mean power along a connectingpath extending from a first point in the distance area to a second pointin the reading area; and specifying mean power over a coordinate systemdistributed over the surface of the lens consistently with the meanpower specified along the connecting path.
 5. The method of claim 4,wherein the coordinate system comprises a set of contour lines, eachcontour line intersecting the connecting path, and wherein specifyingmean power over the coordinate system further comprises specifying meanpower variation along the contour lines as a function, the mean power ona contour line and the mean power on the connecting path being equal atevery point where a contour line intersects the connecting path.
 6. Themethod of claim 5, further comprising rotating in a controlled mannerthe specified mean power values with respect to the plurality of pointsdistributed over the lens surface and the specified lens edge heightwith respect to the edge of the lens.
 7. The method of claim 1, whereinthe step of determining the lens height at a plurality of pointscomprises: defining a mesh comprising a plurality of points over thesurface of the lens; determining mean power at each point on the mesh asdefined by the specified mean power distribution over the lens surface;and numerically solving on the mesh a partial differential equation ofthe elliptic type, subject to the lens edge height as a boundarycondition, to determine the lens height at each point of the mesh. 8.The method of claim 7, wherein the lens comprises a distance area and areading area, and wherein the step of specifying mean power furthercomprises: specifying mean power along a connecting path extending froma first point in the distance area to a second point in the readingarea; and specifying mean power over a coordinate system widelydistributed over the entire area of the lens consistently with the meanpower specified along the connecting path.
 9. The method of claim 8,wherein the coordinate system comprises a set of contour lines, eachcontour line intersecting the connecting path, and wherein specifyingmean power over the coordinate system further comprises specifying meanpower variation along the contour lines as a function, the mean power ona contour line and the mean power on the connecting path being equal atevery point where a contour line intersects the connecting path.
 10. Themethod of claim 9, further comprising rotating in a controlled mannerthe specified mean power values with respect to the plurality of pointsdistributed over the lens surface and the specified lens edge heightwith respect to the edge of the lens.
 11. The method of claim 1, whereinthe lens comprises a distance area and a reading area, and wherein thestep of specifying mean power further comprises: specifying mean poweralong a connecting path extending from a first point in the distancearea to a second point in the reading area; and specifying mean powerover a coordinate system distributed over the surface of the lensconsistently with the mean power specified along the connecting path.12. The method of claim 11, wherein the step of specifying mean powerover the coordinate system further comprises: specifying mean power inthe distance area and reading area; and specifying mean power over acoordinate system distributed over the remaining area of the lensconsistently with the mean power specified along the connecting path.13. The method of claim 11, wherein the coordinate system comprises aset of contour lines, each contour line intersecting the connectingpath, and wherein specifying mean power over the coordinate systemfurther comprises specifying mean power variation along the contourlines as a function, the mean power on a contour line and the mean poweron the connecting path being equal at every point where a contour lineintersects the connecting path.
 14. The method of claim 13, wherein eachcontour line intersects the connecting path only once and each contourline does not intersect any other contour line.
 15. The method of claim14, wherein the mean power is constant along the length of each contourline.
 16. The method of claim 14, wherein the mean power varies alongthe length of each contour line.
 17. The method of claim 11, whereinspecifying the lens height around the edge of the lens comprisesdefining a boundary height profile function in which the boundary heightvaries only slightly in a first boundary segment adjacent to thedistance area and a second boundary segment adjacent to the readingarea, and the boundary height undergoes substantially smooth transitionsbetween the first and second boundary segments.
 18. The method of claim11, further comprising redistributing astigmatism away from theconnecting path.
 19. The method of claim 18, wherein the step ofredistributing astigmatism comprises: determining a change in mean powerrequired to reduce astigmatism on the connecting path; distributing thechange in mean power across the lens to modify the specified mean powervalues; determining lens height at the points of the plurality of pointsdistributed over the lens surface using the modified mean power valuesat the points.
 20. The method of claim 1, further comprising rotating ina controlled manner the specified mean power values with respect to theplurality of points distributed over the lens surface and the specifiedlens edge height with respect to the edge of the lens.
 21. The method ofclaim 20, wherein the rotation is controlled by an angle-dependenthanding function H(θ) such that M(ρ,θ)→M(ρ,H(θ)) and z(θ)→z(H(θ)) whereM is mean power, z(θ) is the lens edge height, and (ρ,θ) are polarcoordinates over the area of the lens.
 22. A progressive lens comprisinga surface having a variable height and including a distance area and areading area, wherein mean power over the lens surface varies accordingto a set of curves forming iso-mean power contours on the lens surfaceand a contour defining an area of constant mean power in the distancearea that is substantially elliptical, and wherein mean power M variesalong a connecting path extending from a first point in the distancearea to a second point in the reading area according to a function ofthe form:${M(y)} = {M_{D} + {\left\lbrack \frac{M_{R} - M_{D}}{2} \right\rbrack\left\lbrack {1 - {\cos \left( {\pi \quad \frac{y_{D} - y}{y_{D} - y_{R}}} \right)}} \right\rbrack}}$

where M_(D) is mean power specified at the a first point (0,y_(D)) inthe distance area and M_(R) is mean power specified at the second point(0,y_(R)) in the reading area.
 23. The progressive lens of claim 22,where astigmatism along the connecting path is less than0.15*(M_(R)−M_(D)) where M_(D) is mean power specified at the firstpoint in the distance area and M_(R) is mean power specified at thesecond point in the reading area.
 24. A system for manufacturing aprogressive lens, the system comprising: a processor for acceptinginputs defining mean power variation over a coordinate system coveringthe surface of the lens and defining lens height around the edge of thelens and for calculating lens height at a plurality of points over thelens surface by solving an elliptic partial differential equationsubject to the lens height at the edge of the lens as a boundarycondition; and a memory for storing the calculated lens height values.25. The system of claim 24, further comprising a numerically controlledmilling machine for accepting the calculated lens height values andusing the calculated lens height values for machining the lens or a moldfor producing the lens or a former for producing the lens.
 26. Thesystem of claim 25, wherein the numerically controlled milling machineadjusts the calculated lens height values in accordance with the type ofmachining tool fitted to the milling machine.
 27. A system formanufacturing a progressive lens, the system comprising: means foraccepting inputs defining mean power variation over a coordinate systemcovering the surface of the lens and defining lens height around theedge of the lens and for calculating lens height at a plurality ofpoints over the lens surface by solving an elliptic partial differentialequation subject to the lens height at the edge of the lens as aboundary condition; and means for storing the calculated lens heightvalues.
 28. The system of claim 27, further comprising means foraccepting the calculated lens height values and using the calculatedlens height values for machining the lens or a mold for producing thelens or a former for producing the lens.
 29. A system for designing aprogressive lens comprising a processor for calculating lens height at aplurality of points over the lens surface according to the method ofclaim
 1. 30. A system for designing progressive lens, the systemcomprising: a processor for accepting inputs defining mean powervariation over a coordinate system covering the surface of the lens anddefining lens height around the edge of the lens and for calculatinglens height at a plurality of points over the lens surface by solving anelliptic partial differential equation subject to the lens height at theedge of the lens as a boundary condition; and a memory for storing thecalculated lens height values.
 31. A system for designing a progressivelens comprising: means for accepting inputs defining mean powervariation over a coordinate system covering the surface of the lens anddefining lens height around the edge of the lens and for calculatinglens height at a plurality of points over the lens surface by solving anelliptic partial differential equation subject to the lens height at theedge of the lens as a boundary condition; and means for storing thecalculated lens height values.
 32. A progressive lens comprising asurface having a variable height and including a distance area and areading area, wherein mean power over the lens surface varies accordingto a set of curves forming iso-mean power contours on the lens surfaceand a contour defining an area of constant mean power in the distancearea that is substantially elliptical, and where astigmatism along theconnecting path is less than 0.15* (M_(R)−M_(D)) where M_(D) is meanpower specified at a first point (0,y_(D)) in the distance area andM_(R) is mean power specified at a second point (0,y_(R)) in the readingarea.